Thompson-like characterizations of the solvable radical (original) (raw)

Thompson-like characterization of the solvable radical

Journal of Algebra - J ALGEBRA, 2005

We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.

Characterization of Solvable Groups and Solvable Radical

International Journal of Algebra and Computation, 2013

We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for Lie algebras. Some open problems are discussed as well.

From Thompson to Baer–Suzuki: A sharp characterization of the solvable radical

Journal of Algebra, 2010

We prove that an element g of prime order > 3 belongs to the solvable radical R(G) of a finite (or, more generally, a linear) group if and only if for every x ∈ G the subgroup generated by g, xgx -1 is solvable. This theorem implies that a finite (or a linear) group G is solvable if and only if in each conjugacy class of G every two elements generate a solvable subgroup.

A description of Baer–Suzuki type of the solvable radical of a finite group

Journal of Pure and Applied Algebra, 2009

We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g ∈ G such that for any 3 elements a 1 , a 2 , a 3 ∈ G the subgroup generated by the elements g, a i ga −1 i , i = 1, 2, 3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2, 3}-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups).

Engel-like characterization of radicals in finite dimensional Lie algebras and finite groups

manuscripta mathematica, 2006

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y ∈ G such that for any x ∈ G the nth commutator [x, y, . . . , y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group.

Characterizations of the solvable radical

Proceedings of the American Mathematical Society, 2010

We prove that there exists a constant k k with the property: if C \mathcal {C} is a conjugacy class of a finite group G G such that every k k elements of C \mathcal {C} generate a solvable subgroup, then C \mathcal {C} generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4 k=4 . We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10 k=10 . By lengthening one of our arguments slightly, we obtain a value of k = 7 k=7 .

A commutator description of the solvable radical of a finite group

2006

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of g ∈ G such that for any k elements a1, a2,..., ak ∈ G the subgroup generated by the elements g, aiga −1 i, i = 1,...,k, is solvable. We consider a similar problem of finding the smallest integer ℓ> 1 with the property that R(G) coincides with the collection of g ∈ G such that for any ℓ elements b1, b2,..., bℓ ∈ G the subgroup generated by the commutators [g, bi], i = 1,..., ℓ, is solvable. Conjecturally, k = ℓ = 3. We prove that both k and ℓ are at most 7. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 8 elements generate a

New Trends in Characterization of Solvable Groups

webdoc.sub.gwdg.de

Abstract. We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for ...

On the number of conjugates defining the solvable radical of a finite group

Comptes Rendus Mathematique, 2006

We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) consists of the elements g such that for any k elements a 1 , a 2 ,. .. , a k ∈ G the subgroup generated by the elements g, a i ga −1

Baer–Suzuki theorem for the solvable radical of a finite group

Comptes Rendus Mathematique, 2009

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: